Cumulative elevation gain
Encyclopedia
In running
, cycling
, and mountaineering
, cumulative elevation gain refers to the sum of every gain in elevation
throughout an entire trip. It is sometimes also known as cumulative gain or elevation gain, or often in the context of mountain travel, simply gain. Elevation losses are not counted in this measure. Cumulative elevation gain, along with round-trip distance, is arguably the most important value used in quantifying the strenuousness of a trip. This is because hiking 10 miles (16.1 km) on flat land (zero elevation gain) is significantly easier than hiking up a large mountain with a round-trip distance of 10 miles (16.1 km). It is much harder to ascend vertically, or to increase elevation, than to walk on flat land because doing so also requires that the hiker increase his/her gravitational potential energy.
However, when climbing a mountain with some "ups-and-downs", or traversing several mountains, you must take into account every "up" along the whole route. This even means that the (usually small) uphills on the descent must be counted. For example, consider a mountain whose summit was 5000 feet (1,524 m) in elevation, but somewhere on the way up, the trail went back down 250 feet (76.2 m). If starting at an elevation of 1000 feet (304.8 m), one would gain 4250 feet (1,295.4 m) on the way up (not 4000, because 250 is lost and has to be "regained") and 250 more feet on the way down, leading to a cumulative elevation gain of 4500 feet (1,371.6 m) on the trip.
If one were to hike over five hills of 100 vertical feet each, and back, the cumulative elevation gain would be 5 x (100 ft) x 2 = 1000 ft.
This concept makes travel on mountains which have more "ups-and-downs", or are generally more rugged, significantly more strenuous.
at a linear distance from the starting point 
can be expressed as an elevation profile function
. For instance, the hiker's elevation 4 miles (6.4 km) into the hike would be given by 
.
The derivative
of the elevation profile
is related to the steepness at a distance 
into the trip. We can then define a gain function as the elevation gained cumulatively up to the point 
in a trip. The gain function can be expressed as
,
where
is the Heaviside step function
and the prime mark
is Lagrange's notation for a derivative
. The purpose of the step function is to zero-out all downhills. That is, any downhill along the way would result in a negative value of
, which would make the step function, and consequently the integrand, zero (so that downhills do not get added). A cumulative elevation loss function could easily be constructed by the insertion of a negative sign inside the step function, so that only downhills are factored in.
If the hiker traveled a total distance of
, the cumulative gain on the trip would be given by
.
Running
Running is a means of terrestrial locomotion allowing humans and other animals to move rapidly on foot. It is simply defined in athletics terms as a gait in which at regular points during the running cycle both feet are off the ground...
, cycling
Cycling
Cycling, also called bicycling or biking, is the use of bicycles for transport, recreation, or for sport. Persons engaged in cycling are cyclists or bicyclists...
, and mountaineering
Mountaineering
Mountaineering or mountain climbing is the sport, hobby or profession of hiking, skiing, and climbing mountains. While mountaineering began as attempts to reach the highest point of unclimbed mountains it has branched into specialisations that address different aspects of the mountain and consists...
, cumulative elevation gain refers to the sum of every gain in elevation
Elevation
The elevation of a geographic location is its height above a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface ....
throughout an entire trip. It is sometimes also known as cumulative gain or elevation gain, or often in the context of mountain travel, simply gain. Elevation losses are not counted in this measure. Cumulative elevation gain, along with round-trip distance, is arguably the most important value used in quantifying the strenuousness of a trip. This is because hiking 10 miles (16.1 km) on flat land (zero elevation gain) is significantly easier than hiking up a large mountain with a round-trip distance of 10 miles (16.1 km). It is much harder to ascend vertically, or to increase elevation, than to walk on flat land because doing so also requires that the hiker increase his/her gravitational potential energy.
Computation (simple explanation)
In the simplest case of a trip where hikers only travel up on their way to a single summit, the cumulative elevation gain is simply given by the difference in the summit elevation and the starting elevation. For example, if one were to start hiking at a trailhead with elevation 1000 feet (304.8 m), and hike up to a summit of 5000 feet (1,524 m), the cumulative elevation gain would just be 5000 ft - 1000 ft = 4000 ft. The loss of elevation on the descent is not relevant, because only increases in elevation are considered in this measure.However, when climbing a mountain with some "ups-and-downs", or traversing several mountains, you must take into account every "up" along the whole route. This even means that the (usually small) uphills on the descent must be counted. For example, consider a mountain whose summit was 5000 feet (1,524 m) in elevation, but somewhere on the way up, the trail went back down 250 feet (76.2 m). If starting at an elevation of 1000 feet (304.8 m), one would gain 4250 feet (1,295.4 m) on the way up (not 4000, because 250 is lost and has to be "regained") and 250 more feet on the way down, leading to a cumulative elevation gain of 4500 feet (1,371.6 m) on the trip.
If one were to hike over five hills of 100 vertical feet each, and back, the cumulative elevation gain would be 5 x (100 ft) x 2 = 1000 ft.
This concept makes travel on mountains which have more "ups-and-downs", or are generally more rugged, significantly more strenuous.
Computation (mathematical explanation)
More generally, suppose a hiker's elevation at a linear distance from the starting point can be expressed as an elevation profile functionFunction (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
. For instance, the hiker's elevation 4 miles (6.4 km) into the hike would be given by .The derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of the elevation profile
is related to the steepness at a distance into the trip. We can then define a gain function as the elevation gained cumulatively up to the point in a trip. The gain function can be expressed as,where
is the Heaviside step functionHeaviside step function
The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....
and the prime mark
Prime (symbol)
The prime symbol , double prime symbol , and triple prime symbol , etc., are used to designate several different units, and for various other purposes in mathematics, the sciences and linguistics...
is Lagrange's notation for a derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
. The purpose of the step function is to zero-out all downhills. That is, any downhill along the way would result in a negative value of
, which would make the step function, and consequently the integrand, zero (so that downhills do not get added). A cumulative elevation loss function could easily be constructed by the insertion of a negative sign inside the step function, so that only downhills are factored in.If the hiker traveled a total distance of
, the cumulative gain on the trip would be given by.